Dedicated to one of the basic concepts of mathematical analysis - the limit. Both in the case of a numerical sequence and in the case of a real function of a real variable, an unlimited approximation to a certain constant value of a variable is investigated, depending on another variable with a certain change in it. In this chapter, we will try to generalize the concept of a limit for mappings of arbitrary metric spaces, and the generalization will also touch on the way the independent variable tends to a given value. 8.1. The concept of the limit of a mapping Let X and Y be metric spaces with the metrics p and d given on them, respectively, X be some subset in X with the same metric /> having a 6 X as its limit point. We emphasize that, by Definition 5.9, this limiting point for A may or may not belong to the subset A. We will consider the THEORY OF LIMITS. The concept of the limit of the map of a punctured neighborhood U (a) = U (a) \ (a) of a given point. Let the domain of definition of the mapping f: AY include the set A. Note that for a point a this mapping may not be defined. Definition 8.1. The point 6 € Y is called the limit of the map /: A -f Y at a point a along the set A and we write b = lim f (x) or f (x) -> b as x -> a, if, whatever may be a neighborhood V (6) of the point 6, there exists a punctured neighborhood U (a) of the point a in X such that its image for any point x ∈ Ua) PL belongs to Y (6), that is, When (8.1) is satisfied, it is also said that the function f (x) tends to b as x tends along the set A to the point a. Definition 8.1 is fairly general. Depending on what sets are X, Y, ACX and what is the point a € X, you can get different concretizations of this definition. Recall (see 5.2) that any neighborhood of a point includes an e-neighborhood of this point, and any ^ -neighborhood is a neighborhood. Therefore, replacing in (8.1) an arbitrary neighborhood V (6) of the point b ∈ Y by its ^ -neighborhood and the punctured neighborhood of the point a ∈ X by its punctured -neighborhood, we arrive at the following symbolic notation for the definition of the limit of a mapping, which is equivalent to Definition 8.1: For Y With R from (8.1) follows a symbolic notation for the definition of the display limit /: (the limit of the real function):. If 6 = 0 in (8.5)) then the function f (x) is called infinitesimal as x tends to the point a ∈ X along the set A and is written When Y C R we can talk about the infinite limits of the mapping if the point 6 is one of the infinite points (+ oo or -oo) extended number line R or their union (oo). In this case, the neighborhood of each of the listed points, when choosing an arbitrary M> O, will take the form Then from (8.1) three quite similar entries in the symbolic form of definitions of the infinite limits of the function follow:. Example 8.1. Let us show that lim f (x) = c if the mapping / at points of the set A takes the same value c. Indeed, whatever the neighborhood is, THEORY OF LIMITS. The notion of the limit of the mapping V (c) of the point c) Vx to U (a) A / (x) = c, since xe A. Therefore, / (U (a) A) = c € V (c), which corresponds to Definition 8.1. Let us verify that lim / (x) = a if the mapping / is identical, that is, / (x) = x Vx 6 A. In this case, for any neighborhood V (a), choosing U (a) = V (a) \ (a) for the identity map, we obtain that corresponds to (8.1). In particular, when A = R and a corresponds to the infinite point + oo of the extended number line, we have: f (x) -f oo as x + oo. Indeed, for an arbitrary M> 0, as a punctured neighborhood of the infinite point + oo, it suffices to choose the set U (+ oo) = (s € R: x> M) in order to obtain f (x)> M and satisfy condition (8.7). # If in Definition 8.1 X = Y = R and the subset A = (a: € R: x> a), then we arrive at the notion of a right-sided limit of a real function of a real variable at a point a, denoted in 7.2 lim fix). If X = Y = R Note that the set A may coincide with the entire set X. For X = Y = R, this case in Definition 8.1 corresponds to the notion of a two-sided limit of a real function of a real variable, and (if there is no threat of confusion) instead of lim / ( x) just write lim / (x). Of course, speaking of lim / (x), one can consider all possible conceivable subsets of A, but this does not always lead to meaningful non-trivial results. So, if the Dirichlet function is considered on the subset Q С R of rational numbers, then we simply obtain a constant function, the limit of which is established in Example 8.1. In definition 8.1 will lead to the notion of the limit of a sequence of points of an arbitrary metric space Y. In this connection, we give the following definition. Definition 8.2. The point 6 € Y is called the limit of a sequence (yn) of points yn of the metric space Y if, whatever the neighborhood V (6) C Y of point 6, there exists a natural number N such that starting from the number N +1 all points of this sequence fall into this neighborhood, i.e. THEORY OF LIMITS. The concept of the limit of a mapping When (8.10) is satisfied, it is also said that (yn) tends to point 6. Using in (8.10), instead of an arbitrary neighborhood of the point 6, its arbitrary ^ -neighborhood, we will have Comparing (8.11) with (6.28) and Definition 6.5, we conclude that the sequence (yn) of points in the metric space tends to point 6 if the numerical sequence ( d (yn> 6)) distances d (yni b) € R is infinitesimal, that is, In other words, the study of the behavior of sequences of points of an arbitrary metric space is based on the study of the convergence of numerical sequences. Moreover, the limit of a mapping of arbitrary metric spaces is closely related to the limit of sequences. This connection is established by the following theorem. Theorem 8.1. The mapping /: Y has a point 6 € Y as its limit as x tends to a point a along the set A if and only if, under the mapping /, the image of any sequence of points from A tending to a is a sequence of points from Y tending to 6, i.e., e. Suppose that the point 6 6 Y satisfies Definition 8.1 of the limit of the map and (xn) is an arbitrary sequence of points xn from A tending to a point a ∈ X. Then, according to (8.1), whatever the neighborhood V (b) C Y point 6, there exists a punctured neighborhood U (a) C X of point a such that / (u (a) PA) C V (6). By Definition 8.2, starting from some number W + 1 all points of the sequence (xn) tending to a must lie in U (a) nA, ”that is, by virtue of (8.10) Then, starting from the same number, all points f (xn) E Y of the sequence (f (xn)) lie in V (6), which, according to Definition 8.2, means that this sequence tends to 6. To prove the sufficiency of the condition of the theorem, we assume that for any sequence (xn) of points xn from A tending to a, the sequence (f (xn)) of points f (xn) from Y tends to 6. If lim f (x) φ 6, then this would mean the existence of such a number e> 0 that for any choice of 8> 0 there is a point x ∈ A satisfying the conditions p (x, a) and d (f (x) y 6)> e. For arbitrarily small S> О it is possible to indicate a natural number N) such that 1 / N. Then, for each number n> N, there is at least one point from A, which we denote by xn, such that p (xn, ^ Thus, the sequence (xn) composed of such points xn e Ay by virtue of (8.11) tends to a , while (f (xn)) does not tend to 6, which contradicts the original assumption. The resulting contradiction proves the sufficiency of the condition of the theorem. This theorem allows us to formulate a definition equivalent to Definition 8.1. Definition 8.3. A point δ € Y is called the limit of the mapping /: A -> Y at a point a along the set A if, under the mapping /, the image of any sequence of points from A tending to a is a sequence of points from Y tending to b. The symbolic forms of writing this definition and Theorem 8.1 coincide. Example 8.2. Let X = R, A = R, a = + oo and in the mapping /: R R f (x) = cos2 Vx 6 R. Let us show that lim f (x) = lim cos a; does not exist. Take the sequence (a: n) = (2nm), which tends to + oo. Then cosin = cos2nmr = 1, and by virtue of (6.9) lim (cos xn) = 1. But if we take the sequence (xn) = ((2n + 1) n / 2), also tending to + oo, then its image converges to zero. This is contrary to 8.3 definition of the display limit, i.e. the above limit does not exist. Consideration of the sequences (2n (-1) n7r) and ((2n + 1) (- 1) ntr / 2) tending to oo leads to the same conclusion. Note that if we denote then it is legal to write lim cosx = 1 and limcoex = 0. # By comparing Definitions 8.1 and 5.13, the following theorem can be proved. Theorem 8.2. The mapping /: X - + Y will be continuous at the point a € X if and only if the limit of the mapping as x tends to the set X to the point a coincides with the value of / (a), that is, when A Let the mapping / be continuous at a point a in X. Then, by Definition 5.13 of a continuous mapping, whatever the neighborhood V (6) of the point 6 = f (a) € Y, there exists a neighborhood U (a) of the point a € A ) that / (U (a)) С V (6), and LIMIT THEORY. The notion of the limit of a mapping, therefore, there also exists a punctured neighborhood U (a) of the point a, such that / (U (a)) C V (b). According to Definition 8.1, this means that (8.12) is true. Conversely, let (8.12) be satisfied. Then, by Definition 8.1, for any neighborhood V (b) of the point b = f (a) there exists a punctured neighborhood U (a) of a such that f (U (a)) C V (6). Consider a neighborhood U (a) = U (a) U (a). Since / (a) G V (6), according to the properties of the mapping of sets (see 2.1), we have 4 that is the mapping /, by Definition 5.13, is continuous at the point aeX. In view of Theorem 8.2, we can formulate a definition equivalent to Definition 5.13. Definition 8.4. The mapping /: is called continuous at the point a 6 Xy if (8.12) is true. Taking into account Theorems 8.1 and 8.2, we obtain the following statement. Statement 8.1. For the mapping /: X -Y Y to be continuous at the limit point abX, it is necessary and sufficient that the image under the mapping / of any sequence of points from X tending to a is a sequence of points from Y converging to the point / (a). 8.2. Some properties of the limit of a mapping Let X and Y, as in 8.1, be metric spaces, AC X and a € X a limit point of A. Theorem 8.3. If, as x tends to the set A to the point a, the mapping /: X Y has a limit, then it is unique. Suppose that for x -> a the mapping / has two limits 6i and 62, and 61 = 62. Then, when choosing disjoint neighborhoods of these points (V (61) flV (62) = 0), by Definition 8.1, the point a has a punctured neighborhood U (a) such that and, but this is impossible by Definition 2.1 of the map. Theorem 8.4 (on the limit of composition). If there are limits of the maps f: AC X and g: Y Z, and ((x) φb for r -> a, where Xy Y and Z are metric limit spaces for A C X and f (A) C Y, respectively, then exists for x -> a and the limit of the composition (of a composite function) .Choose an arbitrary neighborhood W (c) of the point c. Then, by Definition 8.1 of the limit of a map, one can always find a punctured neighborhood V (6) of 6 such that g (V (6) N f)
How to set up smartphones and PCs. Informational portal
